Surface critical exponents of self-avoiding walks on a square lattice with an adsorbing linear boundary: A computer simulation study.

Abstract

Using the scanning simulation method, we study a model of a single self-avoiding walk (SAW) terminally attached to an adsorbing impenetrable linear boundary on a square lattice; an interaction energy \ensuremath{\varepsilon} (\ensuremath{\varepsilon}0) is defined between the ``surface'' and a step (bond) that lies on the surface. SAW's of up to N=260 steps are studied from samples generated with different values of the scanning parameter, b=3 and 5. In most cases the different samples lead to the same results, which suggests that they are statistically reliable. At the ordinary point (infinite temperature T) our result for the growth parameter, \ensuremath{\mu}=2.638 16\ifmmode\pm\else\textpm\fi{}0.000 02, is equal, within the error bars, to the best known estimate of Enting and Guttmann [J. Phys. A 18, 1007 (1985)]. Also, our value ${\ensuremath{\gamma}}_{1}$=0.9551\ifmmode\pm\else\textpm\fi{}0.0003 agrees very well with Cardy's value ${\ensuremath{\gamma}}_{1}$=61/64=0.953 . . . , obtained from conformal invariance [Nucl. Phys. B 240, 514 (1984)]. At the special point, we obtain independently the estimates ${\ensuremath{\gamma}}_{1}$=1.478\ifmmode\pm\else\textpm\fi{}0.020 and ${\ensuremath{\gamma}}_{11}$=0.860\ifmmode\pm\else\textpm\fi{}0.026 and, therefore, also two independent estimates for \ensuremath{\mu} that are found to be equal and very close to the Enting-Guttmann value. These results for ${\ensuremath{\gamma}}_{1}$ and ${\ensuremath{\gamma}}_{11}$ satisfy the Barber scaling relation. However, our adsorption critical temperature -\ensuremath{\varepsilon}/${\mathit{k}}_{\mathit{B}}$${\mathit{T}}^{\mathrm{*}}$=${\mathit{K}}^{\mathrm{*}}$=0.722\ifmmode\pm\else\textpm\fi{}0.004 is larger than estimates previously obtained by the transfer-matrix method. Correspondingly, our result for the crossover exponent \ensuremath{\varphi}=0.562\ifmmode\pm\else\textpm\fi{}0.020 is significantly larger than a theoretical value of Burkhardt, Eisenriegler, and Guim [Nucl. Phys. B 316, 559 (1989)], \ensuremath{\varphi}=1/2.